It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a k such that the spectral radius satisfies r(AB)...

We consider those positive integers that are not representable as linear combinations of terms of a generalized arithmetic progression with nonnegative integer coefficients. To do this, we make use of the numerical semigroup generated by a generalized arithmetic progression. The number of integers nonrepresentable by such a numerical semigroup is determined as well as that of its dual. In addit...

We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a most general (group-valued or semigroup-v...

In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the imaginary axis and the norm of its resolvent operator is polynomially bounded near these points. We characterize classes of perturbations preserving the strong stabi...

Let T X be the full transformation semigroup on the setX and let T X,Y {α ∈ T X : Xα ⊆ Y}. Then T X,Y is a sub-semigroup of T X determined by a nonempty subset Y of X. In this paper, we give a necessary and sufficient condition for T X,Y to be regular. In the case that T X,Y is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green’s re...

The continuous symmetry methodology has been developed to provide a quantitative estimation for the degree of a selected symmetry point-group in a molecular structure. Previous developed measures included the C(S), C(n), S(n) (including C(i)) symmetries, several polyhedral symmetries, and the related chirality measure. Motivated by the abundance and importance of C(nv) symmetry in many key mole...

Extensions to classical Octrees that add new types of terminal nodes have been proposed for the exact representation of polyhedral objects. In this work, we present a new solid representation scheme using Octrees, which include boundary information of the represented object in the internal nodes of the tree. In this way, basic operations with the model will be accelerated and we are able to rep...